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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A new proof of Sárközy's theorem


Author: Neil Lyall
Journal: Proc. Amer. Math. Soc. 141 (2013), 2253-2264
MSC (2010): Primary 11B30
Published electronically: March 14, 2013
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Abstract: It is a striking and elegant fact (proved independently by Furstenberg and Sárközy) that in any subset of the natural numbers of positive upper density there necessarily exist two distinct elements whose difference is given by a perfect square. In this article we present a new and simple proof of this result by adapting an argument originally developed by Croot and Sisask to give a new proof of Roth's theorem.


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Additional Information

Neil Lyall
Affiliation: Department of Mathematics, The University of Georgia, Athens, Georgia 30602
Email: lyall@math.uga.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11628-X
PII: S 0002-9939(2013)11628-X
Received by editor(s): October 18, 2011
Published electronically: March 14, 2013
Dedicated: Dedicated to Steve Wainger on the occasion of his retirement
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.